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Infinities 2 sequences and series

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9:30 - 11:00 Geometric Sequences 11:30 - 13:00 Sequences, Infinity and ICT 14:00 - 15:30 Quadratic Sequences

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Starter activity Can you make your calculator display the following sequences? Find the 20 th term for each of these sequences.

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Can you find the next two terms of the following sequence 4, 8, 16, 32,....? Geometric Sequences

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Geometric sequences Position number 123456 Sequence48163264128 x2 4, 8, 16, 32,.... A sequence is geometric if where r is a constant called the common ratio x2

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Geometric sequences or geometric progressions, hence the GP notation Different ways to describe this sequence: By listing its first few terms: 4, 8, 16, 32,... By specifying the first term and the common ratio: 1 st term is 4 and common ratio is 2 or By giving its nth term ? By graphical representation ?

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Finding the nth term Position number 12345n Sequence48163264 4x14x24x44x84x16 4x2 0 4x2 1 4x2 2 4x2 3 4x2 4 nth term = 4x2 n-1

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4, 8, 16,... is a divergent sequence

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Geometric sequences Can you find the next two terms of the following sequence? 0.2, 0.02, 0.002,.... Can you describe this sequence in different ways? By listing its terms: By specifying the first term and the common ratio: By finding its nth term: By graphical representation:

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0.2, 0.02, 0.002,... is a convergent sequence The sequence converges to a certain value (or a limit number)

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e.g. it approaches 0 This convergent sequence also oscillates. Another example of a convergent sequence:

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Geometric sequences 1. Can you generate (or find) the first 5 terms of the following GPs? Seq A: Seq B: 2. Can you write down the nth term of these sequences? 3. Are these sequences convergent or divergent? Can you use the limit notation in your answers?

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Geometric sequences 1. What is the ratio and the 7 th term for each of the following GPs? Seq A: 2, 10, 50, 250,...? Seq B: 24,12, 6, 3,....? Seq C: -27, 9, -3, 1,....? Challenge 1 What if you want to find the 50 th term of each of these sequences? How would you change your approach? Challenge 2 The 3 rd term in a geometric sequence is 36 and the 6 th term is 972. What is the value of the 1 st term and the common ratio? Challenge 3 Q6 handout

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Suppose we have a 2 metre length of string...... which we cut in half We leave one half alone and cut the 2 nd in half again... and again cut the last piece in half Geometric Series

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Continuing to cut the end piece in half, we would have in total In theory, we could continue for ever, but the total length would still be 2 metres, so This is an example of an infinite series.

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or is the Greek capital letter S, used for Sum

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Geometric series The sum of all the terms of a geometric sequence is called a geometric series. We can write the sum of the first n terms of a geometric series as: When n is large, how efficient is this method ? S n = a + ar + ar 2 + ar 3 + … + ar n –1 For example, the sum of the first 5 terms of the geometric series with first term 2 and common ratio 3 is: S 5 = 2 + (2 × 3) + (2 × 3 2 ) + (2 × 3 3 ) + (2 × 3 4 ) = 2 + 6 + 18 + 54 + 162 = 242

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The sum of a geometric series Start by writing the sum of the first n terms of a general geometric series with first term a and common ratio r as: Multiplying both sides by r gives: S n = a + ar + ar 2 + ar 3 + … + ar n –1 rS n = ar + ar 2 + ar 3 + … + ar n –1 + ar n Now if we subtract the first equation from the second we have: rS n – S n = ar n – a S n ( r – 1) = a ( r n – 1) Challenge: Can you follow the proof of the formula for the sum of the first n terms of a GS? (in pairs)

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Geometric series a)Find the sum of the first 7 terms of the following GP: 4, - 2, 1,... giving your answer correct to 3 significant figures. b)Calculate: Challenge Is ? What is as an exact fraction?

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